Is it possible that you only have the "valet" key which will not open the trunk? I'm not sure if they even had this on these Corollas. If so, maybe you can go to a Toyota dealer and get them to cut you a new key based on the VIN.
There were valet keys, but they will open the trunk just like the regular key. What they won't do is lock or unlock the trunk.
To lock the trunk, you turn the key to the left instead of the right, and remove the key turned that way rather than returning it to vertical before removing it. To open a locked trunk, you put the key in and turn it 180 degrees to the right. When locked (key slot horizontal), the inside lever is deactivated, and the valet key is blocked from turning the lock. When unlocked (key slot vertical), the valet key can open the trunk.
Having the dealer cut from code is the best way to get a fresh key, and some dealers even do it for free as a goodwill gesture (since our keys are dirt cheap, not having a transponder).
I made my own valet key. The difference between a regular and valet key is just one tumbler at the very tip, so if you borrow a valet key from some other Toyota that uses the same blank (I used one from an '06 Corolla) and just copy that tip cut onto the regular key, then you end up with a valet key.
Never did use that valet key though, but it was a fun exercise to see if I could pull it off.